## Rows: 159
## Columns: 30
## $ NAME <chr> "Rabun", "Towns", "Fannin", "Murray", "Whitf…
## $ FIPS <chr> "13241", "13281", "13111", "13213", "13313",…
## $ county <chr> "241", "281", "111", "213", "313", "047", "2…
## $ rucc_code13 <fct> non, non, non, sm, sm, mm, non, mm, mm, non,…
## $ rucc_code13_n <dbl> 6, 6, 6, 4, 4, 3, 6, 3, 3, 6, 5, 6, 6, 5, 5,…
## $ mortality <dbl> 4, 3, 5, 6, 15, 10, 2, 13, 4, 6, 10, 8, 7, 7…
## $ population <dbl> 16602, 11506, 25322, 39782, 104658, 66550, 2…
## $ Year <dbl> 2020, 2020, 2020, 2020, 2020, 2020, 2020, 20…
## $ state <chr> "13", "13", "13", "13", "13", "13", "13", "1…
## $ pct_poverty <dbl> 13.6, 8.9, 7.6, 11.8, 11.3, 7.2, 12.1, 10.4,…
## $ vacancy_rate <dbl> 44.6, 39.6, 36.1, 10.3, 9.5, 8.8, 31.1, 13.6…
## $ unemployment_rate <dbl> 4.1, 4.2, 5.8, 6.5, 6.0, 3.5, 4.8, 6.8, 5.0,…
## $ unemployment_rate_out <dbl> 4.1, 4.2, 5.8, 6.5, 6.0, 3.5, 4.8, 6.8, 5.0,…
## $ pct_black <dbl> 2.0, 1.8, 0.8, 1.3, 4.5, 3.7, 1.0, 5.2, 2.1,…
## $ dist_to_usroad <dbl> 177343.881, 229218.710, 177629.928, 65168.05…
## $ dist_to_treatment <dbl> 100811.3814, 63782.7232, 79264.3957, 4003.75…
## $ incidence <dbl> 2.409348e-04, 2.607335e-04, 1.974568e-04, 1.…
## $ mort_rate <dbl> 0.24093483, 0.26073353, 0.19745676, 0.150821…
## $ Name <chr> "Clayton", "Hiawassee", "Blue Ridge", "Chats…
## $ Name_Seat <chr> "Clayton", "Hiawassee", "Blue Ridge", "Chats…
## $ pct_poverty_std <dbl> -0.15651142, -0.98484701, -1.21396111, -0.47…
## $ vacancy_rate_std <dbl> 3.08075917, 2.51252486, 2.11476084, -0.81732…
## $ unemployment_rate_std <dbl> -0.74498218, -0.70715012, -0.10183723, 0.162…
## $ unemployment_rate_out_std <dbl> -0.793303461, -0.751995315, -0.091064977, 0.…
## $ pct_black_std <dbl> -1.5609990, -1.5723595, -1.6291621, -1.60076…
## $ dist_to_usroad_std <dbl> 0.92635695, 1.52090721, 0.92963541, -0.35931…
## $ dist_to_treatment_std <dbl> 1.211776760, 0.425387080, 0.754176404, -0.84…
## $ rucc_code13_4 <fct> mi_non, mi_non, mi_non, mm_sm, mm_sm, mm_sm,…
## $ rucc_code13_5 <fct> mi_non, mi_non, mi_non, sm, sm, mm, mi_non, …
## $ geometry <MULTIPOLYGON [US_survey_foot]> MULTIPOLYGON (((88…
| Characteristic | Overall N = 1591 |
Mortality rate
|
p-value2 | ||
|---|---|---|---|---|---|
| Low N = 591 |
Moderate N = 611 |
High N = 391 |
|||
| Rural-Urban Continuum Code | 0.008 | ||||
| Large Central Metro & Large Fringe Metro | 29 (18%) | 5 (8.5%) | 18 (30%) | 6 (15%) | |
| Medium Metro | 15 (9.4%) | 3 (5.1%) | 10 (16%) | 2 (5.1%) | |
| Small Metro | 30 (19%) | 15 (25%) | 8 (13%) | 7 (18%) | |
| Micropolitan & Non-Metro | 85 (53%) | 36 (61%) | 25 (41%) | 24 (62%) | |
| Mortality Count | 3 (1, 8) | 1 (0, 2) | 5 (2, 14) | 6 (4, 9) | <0.001 |
| County Population | 22,736 (11,319, 57,089) | 21,498 (10,343, 43,014) | 27,113 (17,277, 91,600) | 20,533 (12,830, 35,871) | 0.040 |
| Poverty rate | 14.0 (10.1, 18.1) | 16.6 (12.7, 20.0) | 11.7 (8.7, 16.6) | 13.8 (10.0, 17.0) | 0.002 |
| Vacancy rate | 16 (12, 21) | 16 (14, 22) | 14 (10, 19) | 19 (12, 27) | 0.042 |
| Unemployment rate | 5.70 (4.30, 7.10) | 5.80 (4.20, 8.60) | 5.60 (4.70, 6.50) | 5.40 (4.20, 6.60) | 0.5 |
| Percentage of Black Population | 29 (17, 41) | 31 (25, 47) | 25 (12, 36) | 30 (11, 41) | 0.012 |
| Distance to interstate | 83,373 (21,222, 136,712) | 90,371 (47,267, 151,962) | 63,821 (10,797, 105,001) | 95,523 (39,523, 149,321) | 0.035 |
| Unknown | 2 | 1 | 1 | 0 | |
| Distance to treatment | 19,036 (3,692, 83,602) | 14,239 (3,721, 90,642) | 21,738 (4,079, 82,826) | 16,063 (2,889, 83,596) | 0.8 |
| Unknown | 2 | 1 | 1 | 0 | |
| 1 n (%); Median (Q1, Q3) | |||||
| 2 Fisher’s exact test; Kruskal-Wallis rank sum test | |||||
| NAME | incidence |
|---|---|
| Turner | 0.0010049 |
| Tift | 0.0007636 |
| Randolph | 0.0005654 |
| Telfair | 0.0005629 |
| Wilkinson | 0.0005581 |
| NAME | pct_poverty |
|---|---|
| Taylor | 28.6 |
| Taliaferro | 28.5 |
| Terrell | 28.0 |
| Jenkins | 27.5 |
| Seminole | 27.1 |
| NAME | vacancy_rate |
|---|---|
| Quitman | 53.2 |
| Rabun | 44.6 |
| Hancock | 43.3 |
| Towns | 39.6 |
| Clay | 39.2 |
| NAME | unemployment_rate_out |
|---|---|
| Quitman | 13.826 |
| Baker | 13.826 |
| Charlton | 13.700 |
| Long | 12.400 |
| Crisp | 12.000 |
| NAME | pct_black |
|---|---|
| Hancock | 72.7 |
| Clayton | 72.5 |
| Dougherty | 71.2 |
| Randolph | 64.4 |
| Macon | 62.6 |
| NAME | dist_to_usroad |
|---|---|
| Seminole | 446751.5 |
| Miller | 388135.0 |
| Early | 371694.7 |
| Decatur | 364239.2 |
| Bacon | 313541.7 |
| NAME | dist_to_treatment |
|---|---|
| Quitman | 185131.5 |
| Warren | 164829.7 |
| Stewart | 161330.9 |
| Glascock | 160496.1 |
| Randolph | 153486.9 |
\[
y_i|\mu_i \sim \text{Poisson}(\mu_i), \\ where \ \mu_i = E(y_i) =
Var(y_i)
\\\
\\
log(\frac{\mu_i}{pop_i}) = \beta_0 + \beta_1\,poverty\_rate_i +
\beta_2\,vacancy\_rate_i + \beta_3\,unemployment\_rate_i +
\beta_4\,pct\_black_i + \beta_5\,dist\_to\_road_i +
\beta_6\,dist\_to\_treatment_i + \theta_i
\\\
\\
log(\mu_i) = log\,pop_i + \beta_0 + \beta_1\,poverty\_rate_i +
\beta_2\,vacancy\_rate_i + \beta_3\,unemployment\_rate_i +
\beta_4\,pct\_black_i + \beta_5\,dist\_to\_road_i +
\beta_6\,dist\_to\_treatment_i + \theta_i
\\\
\\
\theta_i \sim N(0,\tau^2)
\]
\(y_i\) : mortality count for county i
\(\mu_i\) : expected mortality count for county i
\(log\,pop_i\) : population of county i, used as an offset to adjust for the different population sizes across the counties
\(\beta_0\) : baseline log expected mortality rate
\(\theta_i\) : random intercept for county i, county-specific deviation in baseline log expected mortality rate
\(e^{\beta_1}\) : relative mortality rate change for a one standard deviation increase in the poverty rate
\(e^{\beta_2}\) : relative mortality rate change for a one standard deviation increase in the vacancy rate
\(e^{\beta_3}\) : relative mortality rate change for a one standard deviation increase in the unemployment rate
\(e^{\beta_4}\) : relative mortality rate change for a one standard deviation increase in the percentage of black population
\(e^{\beta_5}\) : relative mortality rate change for a one standard deviation increase in the distance to the interstate
\(e^{\beta_6}\) : relative mortality rate change for a one standard deviation increase in the distance to the treatment center
# Fit the poisson regression model
dat$log_pop = log(dat$population)
fit = glmer(mortality ~ offset(log_pop) + pct_poverty_std + vacancy_rate_std +
unemployment_rate_out_std + pct_black_std + dist_to_usroad_std +
dist_to_treatment_std + (1|county),
family = poisson(link = "log"), data = dat)
summary(fit)
## Generalized linear mixed model fit by maximum likelihood (Laplace
## Approximation) [glmerMod]
## Family: poisson ( log )
## Formula: mortality ~ offset(log_pop) + pct_poverty_std + vacancy_rate_std +
## unemployment_rate_out_std + pct_black_std + dist_to_usroad_std +
## dist_to_treatment_std + (1 | county)
## Data: dat
##
## AIC BIC logLik deviance df.resid
## 753.0 777.5 -368.5 737.0 149
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -1.34923 -0.54256 -0.04798 0.37666 2.74739
##
## Random effects:
## Groups Name Variance Std.Dev.
## county (Intercept) 0.1948 0.4413
## Number of obs: 157, groups: county, 157
##
## Fixed effects:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -8.93104 0.06266 -142.541 <2e-16 ***
## pct_poverty_std 0.01450 0.07978 0.182 0.8558
## vacancy_rate_std 0.15993 0.07527 2.125 0.0336 *
## unemployment_rate_out_std -0.13974 0.07645 -1.828 0.0676 .
## pct_black_std -0.07285 0.06573 -1.108 0.2677
## dist_to_usroad_std -0.17990 0.07779 -2.313 0.0207 *
## dist_to_treatment_std 0.03392 0.07356 0.461 0.6447
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Correlation of Fixed Effects:
## (Intr) pct_p_ vcnc__ unm___ pct_b_ dst_t_s_
## pct_pvrty_s 0.087
## vcncy_rt_st 0.026 -0.320
## unmplymn___ 0.086 -0.189 0.126
## pct_blck_st -0.016 -0.402 0.045 -0.343
## dst_t_srd_s 0.181 -0.247 -0.315 -0.004 0.183
## dst_t_trtm_ 0.200 -0.066 -0.356 -0.080 0.047 0.004
The baseline relative mortality rate is \(e^\hat{\beta_0}\) = \(e^{-8.93}\) = 0.0001.
There exists heterogeneity in baseline mortality
rate with a between-county standard deviation \(\tau\) of 0.44.
So
95% of the counties have baseline mortality rates between \(e^{-8.93 \pm 1.96 \times 0.44}\) =
(0.00006, 0.0003).
There is evidence that mortality rate increases
by approximately 17.4% (\(e^\hat{\beta_2}\) = \(e^{0.16}\) = 1.174) for a one standard
deviation increase in the vacancy
rate.
There is evidence that mortality rate decreases by approximately 16.5% (\(e^\hat{\beta_5}\) = \(e^{-0.180}\) = 0.835) for a one standard deviation increase in the distance to the interstate.